Finite Group Databasebeta

← #413235

Subgroups — #413235

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2060C214yesmaximal
3#20800C212yesmaximal
4#13910C211yesmaximal, characteristic
5#11342C214yesmaximal
6#10851C212yesmaximal
7C4×#343C211yesmaximal, characteristic
8C2×#544C2212yes—
9C2×#428C2218yes—
10C2×C4×M16C2212yes—
11#2857—22no—
12#2659—22no—
13#2552—22no—
14#2551—22no—
15#2550—24no—
16#2316C2212yes—
17C2×#389—22no—
18#2285C2212yes—
19C2×#384C2212yes—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2060—216no—
24#1847—24no—
25#1845—22no—
26#1844—22no—
27C2×#343C2212yes—
28C2×#343C2212yes—
29C2×#339C2218yes—
30#1444—22no—
31C4⋊#95C2212yes—
32C22×M16C2314yes—
33#544—28no—
34#544—44no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×#125—24no—
38C2×C22⋊Q8—24no—
39C2×#117C2314yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2311yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C42D412yes—
48C22×C42C2312yes—
49C22×C42C2311yescharacteristic
50#428—216no—
51#428—48no—
52C2×#107—28no—
53C2×#106—28no—
54C4×M16—24no—
55C4×M16Q818yescomplemented
56C2×#97—22no—
57C2×C4×C8—22no—
58#398—28no—
59#395—28no—
60#391—24no—
61#389—24no—
62#389—24no—
63#389—24no—
64#389—24no—
65#389—48no—
66#384—22no—
67#384—28no—
68#384—22no—
69#384—22no—
70#384D412yes—
71#384D412yes—
72#382D414yes—
73#380—24no—
74#380—22no—
75#380—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4Q814yescomplemented
82C2×#95—21no—
83C2×#95—24no—
84C2×#95D412yes—
85#343D418yes—
86#343Q818yescomplemented
87#339—216no—
88C22×Q8—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C2×M16—24no—
97C2×M16—24no—
98C2×M16—24no—
99C2×M16C2×D418yes—
100C2×M16C2×Q818yes—
101C2×M16C2×Q818yes—
102C22×C8—24no—
103C4⋊Q8—416no—
104C4⋊Q8—84no—
105#125—416no—
106C22⋊Q8—48no—
107C22⋊Q8—48no—
108#117—28no—
109#117—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—48no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4—84no—
136C2×C4⋊C4C2×D414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×Q814yes—
140C2×C22⋊C4—24no—
141C2×C42—24no—
142C2×C42—24no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—44no—
148C2×C42C2×D412yes—
149C2×C42C2×D412yes—
150C2×C42C2×Q814yes—
151C2×C42C2×Q814yes—
152C2×C42C2×Q812yes—
153C2×C42#4112yes—
154#107—416no—
155#106—416no—
156#97—28no—
157C4×C8—28no—
158#95—28no—
159#95—24no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—48no—
164C24C22×D411yescharacteristic
165C2×Q8—48no—
166C2×Q8—84no—
167C2×Q8—84no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4#12011yescharacteristic
195C22×C4#12012yes—
196C22×C4#12011yescharacteristic
197C22×C4C22⋊Q812yes—
198C22×C4C22⋊Q812yes—
199C22×C4#12714yes—
200C22×C4C4⋊Q812yes—
201C22×C4C4⋊Q814yes—
202C22×C4C4⋊Q812yes—
203C22×C4C22×D412yes—
204C22×C4C22×D412yes—
205C22×C4C22×Q812yes—
206C22×C4C22×Q812yes—
207C22×C4C22×Q811yescharacteristic
208C22×C4C2×#4111yescharacteristic
209C22×C4H(5,𝔽2)12yes—
210C22×C42−1+412yes—
211M16—28no—
212M16—28no—
213M16—28no—
214M16C4⋊Q8116yescomplemented
215C2×C8—28no—
216C2×C8—28no—
217C2×C8—28no—
218C4⋊C4—216no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C4⋊C4—48no—
224C4⋊C4—416no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—48no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—816no—
235C4⋊C4—82no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C42—28no—
239C42—48no—
240C42—44no—
241C42—44no—
242C42—82no—
243C42C22⋊Q814yes—
244C42C22⋊Q814yes—
245C42C4⋊Q814yes—
246C42C4⋊Q814yes—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23C2×#12011yescharacteristic
252C23C2×C22⋊Q811yescharacteristic
253C23C2×#12711yescharacteristic
254C23C2×C4⋊Q811yescharacteristic
255C23C2×C4⋊Q811yescharacteristic
256C23#53511yescharacteristic
257C23#53611yescharacteristic
258Q8—168no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—28no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#39512yes—
296C2×C4#39514yes—
297C2×C4#39512yes—
298C2×C4#45312yes—
299C2×C4#45612yes—
300C2×C4C2×C22⋊Q812yes—
301C2×C4C2×C4⋊Q814yes—
302C2×C4C2×C4⋊Q812yes—
303C2×C4#54414yes—
304C2×C4D4214yes—
305C2×C4#54712yes—
306C2×C4D4×Q814yes—
307C2×C4D4×Q812yes—
308C2×C4#55412yes—
309C2×C4Q8212yes—
310C8—216no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22C2×#39511yescharacteristic
324C22#243311yescharacteristic
325C22#243411yescharacteristic
326C22#254211yescharacteristic
327C22#256711yescharacteristic
328C22#266011yescharacteristic
329C22#286811yescharacteristic
330C22C2×#45311yescharacteristic
331C22C2×#45611yescharacteristic
332C22#302111yescharacteristic
333C22#302211yescharacteristic
334C4—22no—
335C4—24no—
336C4—48no—
337C4—44no—
338C4—44no—
339C4—84no—
340C4—84no—
341C4—162no—
342C4#206014yescomplemented
343C4#244014yes—
344C4#255114yes—
345C2—21no—
346C2—21no—
347C2—21no—
348C2—21no—
349C2#1514311yescharacteristic
350C2C2×#206011yescharacteristic
351C2#2356211yescharacteristic
352C2#2573911yescharacteristic
353C2#2574011yescharacteristic
354C2#2582411yescharacteristic
355C2#2582511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
55. C4×M16Q8semidirect
81. C4×C4⋊C4Q8semidirect
86. #343Q8semidirect
214. M16C4⋊Q8semidirect
342. C4#2060semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#351672
C2×#20602
#208002
#139102
#113422
#108512
C4×#3432